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Time series analysis and forecasting
Published in Amithirigala Widhanelage Jayawardena, Fluid Mechanics, Hydraulics, Hydrology and Water Resources for Civil Engineers, 2021
Amithirigala Widhanelage Jayawardena
The power spectral density function may also be calculated via the fast Fourier transform (FFT) method in which a direct Fourier transform of the original data is carried out. This method is more efficient computationally for series involving large amounts of data. However, when the volume of data is not excessive, and when the auto-correlation function is anyway determined, the FFT method does not offer a great advantage.
Analytical Techniques for Ultra-Wideband Signals
Published in James D. Taylor, Introduction to Ultra-Wideband Radar Systems, 2020
Muriladhar Rangaswamy, Tapan K. Sarkar
We now develop a technique to obtain the power of a signal using its Fourier transform. First we introduce the concept of spectral density. In most of our applications we would be dealing with power signals; hence, it is meaningful to consider only power spectral density in this book. The total area under the spectral density curve plotted as a function of frequency is the average power of the signal. In other words, power spectral density is nothing but the distribution of the average power of the signal as a function of frequency and is expressed in watts/Hertz.
Measurement
Published in Carl Hopkins, Sound Insulation, 2020
White noise is a stationary random signal with constant power spectral density. Gaussian white noise is a particularly useful type of white noise because its statistics are described by the normal (Gaussian) probability distribution. Figure 3.3 shows a record of Gaussian white noise in terms of a positive or negative signal amplitude, x, varying with time. The white noise shown here has a population mean of zero and a population standard deviation of unity. By using a mean of zero, the standard deviation is equal to the rms value of the signal. For an infinite length sample, the distribution of amplitudes is described by the smooth curve of the probability density function, p(x), for a normal distribution. In practice, we use finite length samples of white noise; Fig. 3.3 shows the distributions for two different samples using histograms. The difference between these two histograms serves as a reminder that we will soon need to consider the uncertainty associated with finite length samples when averaging white noise over time in frequency bands. Fortunately, calculation of the statistics for temporal averaging are simplified when the probability density function is known.
Dynamic response optimization of a thermoplastic composite sandwich beam under random vibration
Published in Mechanics of Advanced Materials and Structures, 2023
Akın Oktav, Murat Alper Başaran, Fatih Darıcık
Modal parameters are computed up to 4,000 Hz for VVV, SVS, and SSS models using the Lanczos eigenvalue method. A modal frequency response analysis is then performed using the Sol111 solver of Nastran with 24 steps (i.e. the resolution is 50 Hz) in a bandwidth of 50 Hz-1,250 Hz bandwidth. Through the input location, the beam model is excited with a random signal. The frequency analysis is performed using power spectral density (PSD), which is an ideal tool to describe random vibrations [44]. The power spectral density describes the distribution of power over the excitation frequency range. The input position is assigned a unit acceleration of 1 g in the z-axis direction. The PSD function, tabulated in Table 12, is used as a multiplier for the assigned unit acceleration to generate a random signal [41]. PSD spectrum of random excitation is shown in Figure 8.
Modelling asphalt overlay as-built roughness based on profile transformation. Case for wheeled paver laydown operation
Published in International Journal of Pavement Engineering, 2022
Rodrigo Díaz-Torrealba, José Ramón Marcobal, Juan Gallego
Finally, those works that used assisted levelling during paving were discarded. This information was explicitly available in the construction report for some sections, in other cases it was deduced from spectral density analysis of the longitudinal profiles before and after the overlay. The spectral density is a function that indicates how the variance of the profile is ‘spread out’ over the different sinusoids that, when added together, approximate the profile. In other words, it is a statistical representation of the importance (in terms of amplitude) of the different wavelengths that make up the longitudinal profile. The plot of these amplitudes against their corresponding wavelengths is called Power Spectral Density (PSD) graph or periodogram. The PSD function (ISO 2016) can be computed from both profile elevation and profile slope; on this last case, the resulting amplitudes are more uniform over the wavelengths spectrum, allowing a better visualisation of the different roughness characteristics (Sayers and Karamihas 1998). For the purposes of this study, the spectral density analysis was performed with the PSD plot computed for profile slope using the ProVal® software and considering a 12 octave-band filtering. Examples of the analysis carried out are presented in Figures 8 and 9 for the case with and without assisted levelling.
Effects of basketball court construction and shoe stiffness on countermovement jump landings
Published in Footwear Science, 2019
Olivia L. Bruce, Colin R. Firminger, John W. Wannop, Darren J. Stefanyshyn, W. Brent Edwards
Accelerometer data were analysed in both the time and frequency domains. For the time domain, attenuation index values between the tibia (axial) and head, and between the tibia (resultant) and head were calculated using the equation: where a positive value indicates an attenuation, and a negative value indicates a gain in the acceleration signal. For the frequency domain analysis, data were converted to the frequency domain using a Fast Fourier Transform that calculated the power spectral density (PSD) of the signal from 0 to the Nyquist frequency. The data were split into low (1–15 Hz) and high (16–45 Hz) frequency bands. These frequency bands were identified through examination of the bimodal shape of the PSD curve for the axial tibia accelerometer signal, similar to a previous study (Zhang, Derrick, Evans, & Yu, 2008). Signal power magnitude was calculated as the area under the power spectral density curve within the high-frequency band (Edwards et al., 2012; Gruber et al., 2014). A transfer function was calculated for each frequency using the following equation: where PSD is the power spectral density of the acceleration signal. The transfer function represents a gain (positive values) or attenuation (negative values) in acceleration signal strength between the distal and proximal segments. Transfer function values were converted to a linear scale, averaged within the high-frequency band, and converted back to decibels to obtain a measure of impact attenuation in the frequency domain.